Our function here, which I will call $h(x) = (x^2-1)^2$, is a composition of two functions. You have the inner function which I will define as a $g(x) = x^2 - 1$
and the outer function which takes this inner function g(x) and raises it two the power of two ($(g(x)^2$). Which is a function of itself that I will define as $f(x)$.

The chain rule states $\frac{d}{dx}h(x) = f'(g(x))g'(x)$

In a simplified analogy, you can think about this as multiplying the derivative of the "outside" (keeping the "inside" the same) by the derivative of the inside.

Notice in the first step, all I did was use the power rule on the exponent on the outside of the parentheses and kept the stuff inside the parentheses constant. Then I multiplied this result by the derivative of the stuff on the inside of the parentheses.